A proof theory for machine code

Abstract

This paper develops a proof theory for low-level code languages. We first define a proof system, which we refer to as the sequential sequent calculus, and show that it enjoys the cut elimination property and that its expressive power is the same as that of the natural deduction proof system. We then establish the Curry-Howard isomorphism between this proof system and a low-level code language by showing the following properties: (1) the set of proofs and the set of typed codes is in one-to-one correspondence, (2) the operational semantics of the code language is directly derived from the cut elimination procedure of the proof system, and (3) compilation and de-compilation algorithms between the code language and the typed lambda calculus are extracted from the proof transformations between the sequential sequent calculus and the natural deduction proof system. This logical framework serves as a basis for the development of type systems of various low-level code languages, type-preserving compilation, and static code analysis.

Sequent calculi and abstract machines

“…In our approach they are expressed inside the logic itself. Higuchi and Ohori already have stressed the importance of a Curry-Howard isomorphism for low-level code [Higuchi and Ohori 2002;Ohori 2005]. They propose typing Java bytecode with an extension of intuitionistic propositional calculus.…”

“…Compilation is essentially a proof transformation [Ohori 2005]. It embeds the proof system of Figure 11 into the one of Figure 42, as shown next.…”